Universität Potsdam Nikolai Tarkhanov A Simple Numerical Approach to the Riemann Hypothesis Preprints des Instituts für Mathematik der Universität Potsdam 1 (2012) 9 Preprints des Instituts für Mathematik der Universität Potsdam Preprints des Instituts für Mathematik der Universität Potsdam 1 (2012) 9 Nikolai Tarkhanov A Simple Numerical Approach to the Riemann Hypothesis Universitätsverlag Potsdam Bibliografische Information der Deutschen Nationalbibliothek Die Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über http://dnb.de abrufbar. Universitätsverlag Potsdam 2012 http://info.ub.uni-potsdam.de/verlag.htm Am Neuen Palais 10, 14469 Potsdam Tel.: +49 (0)331 977 2533 / Fax: 2292 E-Mail: [email protected] Die Schriftenreihe Preprints des Instituts für Mathematik der Universität Potsdam wird herausgegeben vom Institut für Mathematik der Universität Potsdam. ISSN (online) 2193-6943 Kontakt: Institut für Mathematik Am Neuen Palais 10 14469 Potsdam Tel.: +49 (0)331 977 1028 WWW: http://www.math.uni-potsdam.de Titelabbildungen: 1. Karla Fritze | Institutsgebäude auf dem Campus Neues Palais 2. Nicolas Curien, Wendelin Werner | Random hyperbolic triangulation Published at: http://arxiv.org/abs/1105.5089 Das Manuskript ist urheberrechtlich geschützt. Online veröffentlicht auf dem Publikationsserver der Universität Potsdam URL http://pub.ub.uni-potsdam.de/volltexte/2012/5764/ URN urn:nbn:de:kobv:517-opus-57645 http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-57645 A SIMPLE NUMERICAL APPROACH TO THE RIEMANN HYPOTHESIS N. TARKHANOV This paper is dedicated to P. M. Gauthier on the occasion of his 70 th birthday Abstract. The Riemann hypothesis is equivalent to the fact that the recip- rocal function 1/ζ(s) extends from the interval (1/2, 1) to an analytic function in the quarter-strip 1/2 < s<1, s>0. Function theory allows one to rewrite the condition of analytic continuability in an elegant form amenable to numerical experiments. Contents Introduction 1 1. The Riemann zeta function 2 2. Analytic continuation in a lune 3 3. A Carleman formula for a half-disk 6 4. Reduction of the Riemann hypothesis 8 5. Numerical experiments 11 References 12 Introduction The Riemann hypothesis is that all zeros of the Riemann zeta function ζ(z) in the critical strip 0 < z<1 belong to the critical line z =1/2. This just amounts to saying that the function 1/ζ(z) extends from the interval (1/2, 1) to an analytic function in the quarter-strip 1/2 < z<1, z>0. Note that the restriction of 1/ζ(s)to(1/2, 1) is actually continuous on the closed interval. Hence the function theory allows one to rewrite the condition of analytic continuability in an elegant form which is amenable to numerical experiments. More precisely, one constructs an explicit sequence {cn} of complex numbers, such that the equality n lim |cn| = 1 is fulfilled if and only if the Riemann hypothesis is true. The numbers cn are integrals of 1/ζ(s) over the interval [1/2, 1] with explicit weight function depending on n. Computations with the newest versions of Mathematica, Maple and Matlab performed by my diploma students give certain evidence to the fact that the limit is 1 indeed. However, the standard computer programmes are not n sufficient to evaluate the sequence |cn| with strict accuracy. The numerical data 2000 Mathematics Subject Classification. Primary 11M26; Secondary 11Mxx. Key words and phrases. Zeta function, Carleman formulas, numerical methods. The author wishes to express his thanks to S. Grudsky for several helpful comments concerning asymptotic formulas. 1 2 N. TARKHANOV n obtained in this way testify to lim |cn| = 1 not only for 1/ζ(s) but also for other functions (e.g. 1/(s − 3/4 − ı/2)) which fail to be analytically extendable to the critical quarter-strip. Thus the paper gives rise to the problem of elaborating an n efficient programme which recognises through the behaviour of the sequence |cn| those continuous functions on [1/2, 1] which extend to analytic functions in the quarter-strip. 1. The Riemann zeta function In this section we gather necessary material about the Riemann zeta function. For complete proofs the reader is referred√ to [Tit51, KV92, Con03]. For complex numbers s = s + −1s in the half-plane s>1 the Riemann function is defined by ∞ 1 ζ(s)= , ns n=1 the series converging absolutely and uniformly in each half-plane s>s0 with s0 > 1. In 1737 Euler proved his product formula which displayed a deep connection of ζ(s) with the distribution of prime numbers. Theorem 1.1. If s>1 then 1 −1 ζ(s)= 1 − , ps p where the product runs over all prime numbers p (p =1is no prime number). In order to extend ζ(s) to an analytic function on all of C, one uses the analytic extension of the gamma function constructed by Weierstraß. More precisely, ∞ 1 1 = seγs 1+ e−s/n Γ(s) n n=1 holds for s>0, where γ is the Euler-Mascheroni constant. The right-hand side of this equality is an entire function of s vanishing at the points s =0, −1, −2,.... Lemma 1.2. If s>1 then s/2 ∞ ∞ π 1 2 ζ(s)= + xs/2−1 + x−s/2−1/2 e−πn x dx . Γ(s/2) s(s − 1) 1 n=1 The lemma shows that the Riemann zeta function extends to a meromorphic function in the whole complex plane with the only pole at s = 1 which is simple. This function vanishes at s = −2, −4,..., the other zeros of ζ(s)areknownto lie in the critical strip 0 < s<1. B. Riemann conjectured (1869) that all zeros of ζ(s) in the critical strip belong to the line s =1/2. The restriction of ζ(s) to the critical strip is symmetric with respect to both the critical line s =1/2andtheinterval(0, 1) of the real axis. Moreover, it is different from zero for all s ∈ [0, 1]. Hence the Riemann hypothesis just amounts to saying that ζ(s) has no zeros in the quarter-strip 1/2 < s<1, s>0. A SIMPLE NUMERICAL APPROACH TO THE RIEMANN HYPOTHESIS 3 For real x>0, let π(x) denote the number of prime numbers p which satisfy p ≤ x. B. Riemann showed a formula for the difference x ds π x − ( ) s 0 log in terms of x and zeros of ζ(s) lying in the critical strip. If ζ(s) has no zeros with s>s0 for some 1/2 ≤ s0 < 1, then the asymptotic formula x ds π x O xs0 x ( )= s + ( log ) 0 log holds. The Riemann hypothesis just amounts to this formula with s0 =1/2. Some textbooks in complex analysis include the so-called prime number theorem proved independently by J. Hadamard and Ch.-J. de la Vall´ee-Poussin (1896). It reads π(x) ∼ x/ log x. 2. Analytic continuation in a lune Denote by D := {w ∈ C : |w| < 1} the open unit disk with center at the origin in the plane of complex variable w.LetS be a regular curve in D,whose endpoints lie on the unit circle and which does not run through 0 (i.e. 0 ∈/ S). The curve S divides the disk D into two domains and we write G for the subdomain of D that does not contain the origin 0. In this way we obtain a bounded domain with piecewiese smooth boundary which is referred to as lune. The boundary of G consists of two parts, one of the two is the curve S and the other an arc of the circle ∂D, see Figure 1. Fig. 1. A basic domain. In 1926 T. Carleman found a simple formula for analytic continuation in a corner. For this reason the following refined formula is named after him. This formula is well known, see for instance [Aiz93]. Since the proof is very simple we give it for completeness. Theorem 2.1. Suppose f is a holomorphic function in G continuous up to the boundary of G.Then w n dw f w f w 1 ( ) = lim ( ) (2.1) n→∞ S w 2πı w − w 4 N. TARKHANOV for all w ∈ G. Proof. Fix an arbitrary w ∈ G.Since0∈/ G and f is holomorphic in G and continuous up to the boundary, the function w n F (w):=f(w) w is holomorphic in G and continuous on G for all n =1, 2,.... By the integral formula of Cauchy one gets dw F w F w 1 ( )= ( ) ∂G 2πı w − w for all w ∈ G. Substituting F yields w n 1 dw f(w)= f(w) . w 2πı w − w ∂G for each n =1, 2,.... The integral on the right-hand side splits into two integrals, the first is over S and the second one over ∂G \ S.So w n dw w n dw f w f w 1 f w 1 . ( )= ( ) + ( ) S w 2πı w − w ∂G\S w 2πı w − w On letting n →∞one obtains w n dw w n dw f w f w 1 f w 1 , ( ) = lim ( ) + lim ( ) n→∞ S w 2πı w − w n→∞ ∂G\S w 2πı w − w in the case when at least one of the limits exists. We now show that the second limit exists and is precisely zero. Since w ∈ G and w ∈ ∂G \ S,weget w |w| = < 1. w 1 It follows that the sequence (w/w)n converges to zero uniformly in w ∈ ∂G \ S. So the limit of the second integral can be evaluated under the integral sign and one obtains w n dw w n dw f w 1 f w 1 lim ( ) = ( ) lim n→∞ ∂G\S w 2πı w − w ∂G\S n→∞ w 2πı w − w =0, which establishes the desired formula.